Derivative of √(2x²-4x+1)

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   ________________
  /    2           
\/  2*x  - 4*x + 1

$$\sqrt{2 x^{2} - 4 x + 1}$$

  /   ________________\
d |  /    2           |
--\\/  2*x  - 4*x + 1 /
dx

$$\frac{d}{d x} \sqrt{2 x^{2} - 4 x + 1}$$

Transform

Detail solution

Let $u = 2 x^{2} - 4 x + 1$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x^{2} - 4 x + 1\right)$ :
1. Differentiate $2 x^{2} - 4 x + 1$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $4 x$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $4$
    So, the result is: $-4$
  3. The derivative of the constant $1$ is zero.
  The result is: $4 x - 4$
The result of the chain rule is:

$\frac{4 x - 4}{2 \sqrt{2 x^{2} - 4 x + 1}}$
Now simplify:

$\frac{2 \left(x - 1\right)}{\sqrt{2 x^{2} - 4 x + 1}}$

The answer is:

$\frac{2 \left(x - 1\right)}{\sqrt{2 x^{2} - 4 x + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      -2 + 2*x     
-------------------
   ________________
  /    2           
\/  2*x  - 4*x + 1

$$\frac{2 x - 2}{\sqrt{2 x^{2} - 4 x + 1}}$$

Simplify

The second derivative [src]

  /               2  \
  |     2*(-1 + x)   |
2*|1 - --------------|
  |                 2|
  \    1 - 4*x + 2*x /
----------------------
    ________________  
   /              2   
 \/  1 - 4*x + 2*x

$$\frac{2 \left(- \frac{2 \left(x - 1\right)^{2}}{2 x^{2} - 4 x + 1} + 1\right)}{\sqrt{2 x^{2} - 4 x + 1}}$$

Simplify

The third derivative [src]

            /                2  \
            |      2*(-1 + x)   |
12*(-1 + x)*|-1 + --------------|
            |                  2|
            \     1 - 4*x + 2*x /
---------------------------------
                       3/2       
       /             2\          
       \1 - 4*x + 2*x /

$$\frac{12 \left(x - 1\right) \left(\frac{2 \left(x - 1\right)^{2}}{2 x^{2} - 4 x + 1} - 1\right)}{\left(2 x^{2} - 4 x + 1\right)^{\frac{3}{2}}}$$

Simplify

The graph

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Derivative of √(2x²-4x+1)

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The solution